LMFDB - Embedded newform 7225.2.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.93924352.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 17x^{4} + 73x^{2} - 67 $ x^6 - 17x^4 + 73x^2 - 67
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.12261$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.311108 q^{2} +1.12261 q^{3} -1.90321 q^{4} +0.349253 q^{6} +3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} -5.74499 q^{11} -2.13656 q^{12} +2.59210 q^{13} +1.12261 q^{14} +3.42864 q^{16} -0.541249 q^{18} -4.28100 q^{19} +4.05086 q^{21} -1.78731 q^{22} +1.47186 q^{23} -1.36321 q^{24} +0.806424 q^{26} -5.32089 q^{27} -6.86760 q^{28} +5.50439 q^{29} +8.33946 q^{31} +3.49532 q^{32} -6.44938 q^{33} +3.31111 q^{36} -6.20290 q^{37} -1.33185 q^{38} +2.90992 q^{39} +3.95768 q^{41} +1.26025 q^{42} -9.76049 q^{43} +10.9339 q^{44} +0.457908 q^{46} +6.62222 q^{47} +3.84902 q^{48} +6.02074 q^{49} -4.93332 q^{52} -0.658781 q^{53} -1.65537 q^{54} -4.38178 q^{56} -4.80589 q^{57} +1.71246 q^{58} -5.95407 q^{59} +8.76357 q^{61} +2.59447 q^{62} -6.27775 q^{63} -5.76986 q^{64} -2.00645 q^{66} -0.428639 q^{67} +1.65233 q^{69} +1.36321 q^{71} +2.11261 q^{72} -3.25917 q^{73} -1.92977 q^{74} +8.14764 q^{76} -20.7304 q^{77} +0.905299 q^{78} +1.89597 q^{79} -0.754037 q^{81} +1.23127 q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} +6.97626 q^{88} -3.52543 q^{89} +9.35342 q^{91} -2.80127 q^{92} +9.36196 q^{93} +2.06022 q^{94} +3.92388 q^{96} +11.7073 q^{97} +1.87310 q^{98} +9.99483 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9} + 2 q^{13} - 6 q^{16} - 16 q^{18} - 12 q^{19} - 2 q^{21} - 22 q^{26} - 6 q^{32} + 28 q^{33} + 20 q^{36} + 32 q^{38} + 34 q^{42} + 8 q^{43} + 40 q^{47} - 4 q^{49}+ \cdots - 16 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 16 * q^9 + 2 * q^13 - 6 * q^16 - 16 * q^18 - 12 * q^19 - 2 * q^21 - 22 * q^26 - 6 * q^32 + 28 * q^33 + 20 * q^36 + 32 * q^38 + 34 * q^42 + 8 * q^43 + 40 * q^47 - 4 * q^49 - 30 * q^52 + 10 * q^53 + 4 * q^59 - 22 * q^64 + 16 * q^66 + 24 * q^67 + 24 * q^69 + 52 * q^72 + 36 * q^76 - 44 * q^77 + 34 * q^81 + 20 * q^83 - 6 * q^84 - 4 * q^86 + 76 * q^87 - 8 * q^89 + 30 * q^93 + 40 * q^94 - 16 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	1.12261	0.648139	0.324070	−	0.946033i	$-0.394949\pi$
$3$	1.12261	0.648139	0.324070	+	0.946033i	$0.394949\pi$
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0.349253	0.142582
$7$	3.60843	1.36386	0.681929	−	0.731419i	$-0.261141\pi$
$7$	3.60843	1.36386	0.681929	+	0.731419i	$0.261141\pi$
$8$	−1.21432	−0.429327
$9$	−1.73975	−0.579916
$10$	0	0
$11$	−5.74499	−1.73218	−0.866090	−	0.499888i	$-0.833374\pi$
$11$	−5.74499	−1.73218	−0.866090	+	0.499888i	$0.833374\pi$
$12$	−2.13656	−0.616773
$13$	2.59210	0.718920	0.359460	−	0.933160i	$-0.382961\pi$
$13$	2.59210	0.718920	0.359460	+	0.933160i	$0.382961\pi$
$14$	1.12261	0.300030
$15$	0	0
$16$	3.42864	0.857160
$17$	0	0
$17$	0	0
$18$	−0.541249	−0.127574
$19$	−4.28100	−0.982128	−0.491064	−	0.871124i	$-0.663392\pi$
$19$	−4.28100	−0.982128	−0.491064	+	0.871124i	$0.663392\pi$
$20$	0	0
$21$	4.05086	0.883969
$22$	−1.78731	−0.381056
$23$	1.47186	0.306905	0.153452	−	0.988156i	$-0.450961\pi$
$23$	1.47186	0.306905	0.153452	+	0.988156i	$0.450961\pi$
$24$	−1.36321	−0.278264
$25$	0	0
$26$	0.806424	0.158153
$27$	−5.32089	−1.02401
$28$	−6.86760	−1.29785
$29$	5.50439	1.02214	0.511070	−	0.859539i	$-0.329249\pi$
$29$	5.50439	1.02214	0.511070	+	0.859539i	$0.329249\pi$
$30$	0	0
$31$	8.33946	1.49781	0.748906	−	0.662676i	$-0.230579\pi$
$31$	8.33946	1.49781	0.748906	+	0.662676i	$0.230579\pi$
$32$	3.49532	0.617890
$33$	−6.44938	−1.12269
$34$	0	0
$35$	0	0
$36$	3.31111	0.551851
$37$	−6.20290	−1.01975	−0.509875	−	0.860248i	$-0.670308\pi$
$37$	−6.20290	−1.01975	−0.509875	+	0.860248i	$0.670308\pi$
$38$	−1.33185	−0.216055
$39$	2.90992	0.465960
$40$	0	0
$41$	3.95768	0.618086	0.309043	−	0.951048i	$-0.399991\pi$
$41$	3.95768	0.618086	0.309043	+	0.951048i	$0.399991\pi$
$42$	1.26025	0.194461
$43$	−9.76049	−1.48846	−0.744230	−	0.667923i	$-0.767184\pi$
$43$	−9.76049	−1.48846	−0.744230	+	0.667923i	$0.767184\pi$
$44$	10.9339	1.64835
$45$	0	0
$46$	0.457908	0.0675148
$47$	6.62222	0.965949	0.482975	−	0.875634i	$-0.339556\pi$
$47$	6.62222	0.965949	0.482975	+	0.875634i	$0.339556\pi$
$48$	3.84902	0.555559
$49$	6.02074	0.860106
$50$	0	0
$51$	0	0
$52$	−4.93332	−0.684129
$53$	−0.658781	−0.0904905	−0.0452452	−	0.998976i	$-0.514407\pi$
$53$	−0.658781	−0.0904905	−0.0452452	+	0.998976i	$0.514407\pi$
$54$	−1.65537	−0.225267
$55$	0	0
$56$	−4.38178	−0.585540
$57$	−4.80589	−0.636555
$58$	1.71246	0.224857
$59$	−5.95407	−0.775154	−0.387577	−	0.921837i	$-0.626688\pi$
$59$	−5.95407	−0.775154	−0.387577	+	0.921837i	$0.626688\pi$
$60$	0	0
$61$	8.76357	1.12206	0.561030	−	0.827796i	$-0.310405\pi$
$61$	8.76357	1.12206	0.561030	+	0.827796i	$0.310405\pi$
$62$	2.59447	0.329498
$63$	−6.27775	−0.790922
$64$	−5.76986	−0.721232
$65$	0	0
$66$	−2.00645	−0.246977
$67$	−0.428639	−0.0523666	−0.0261833	−	0.999657i	$-0.508335\pi$
$67$	−0.428639	−0.0523666	−0.0261833	+	0.999657i	$0.508335\pi$
$68$	0	0
$69$	1.65233	0.198917
$70$	0	0
$71$	1.36321	0.161783	0.0808915	−	0.996723i	$-0.474223\pi$
$71$	1.36321	0.161783	0.0808915	+	0.996723i	$0.474223\pi$
$72$	2.11261	0.248973
$73$	−3.25917	−0.381457	−0.190729	−	0.981643i	$-0.561085\pi$
$73$	−3.25917	−0.381457	−0.190729	+	0.981643i	$0.561085\pi$
$74$	−1.92977	−0.224331
$75$	0	0
$76$	8.14764	0.934599
$77$	−20.7304	−2.36245
$78$	0.905299	0.102505
$79$	1.89597	0.213313	0.106656	−	0.994296i	$-0.465985\pi$
$79$	1.89597	0.213313	0.106656	+	0.994296i	$0.465985\pi$
$80$	0	0
$81$	−0.754037	−0.0837819
$82$	1.23127	0.135970
$83$	16.6637	1.82908	0.914540	−	0.404497i	$-0.132553\pi$
$83$	16.6637	1.82908	0.914540	+	0.404497i	$0.132553\pi$
$84$	−7.70964	−0.841190
$85$	0	0
$86$	−3.03657	−0.327441
$87$	6.17929	0.662489
$88$	6.97626	0.743671
$89$	−3.52543	−0.373695	−0.186847	−	0.982389i	$-0.559827\pi$
$89$	−3.52543	−0.373695	−0.186847	+	0.982389i	$0.559827\pi$
$90$	0	0
$91$	9.35342	0.980505
$92$	−2.80127	−0.292052
$93$	9.36196	0.970790
$94$	2.06022	0.212496
$95$	0	0
$96$	3.92388	0.400479
$97$	11.7073	1.18870	0.594348	−	0.804208i	$-0.297410\pi$
$97$	11.7073	1.18870	0.594348	+	0.804208i	$0.297410\pi$
$98$	1.87310	0.189212
$99$	9.99483	1.00452
$100$	0	0
$101$	3.96989	0.395019	0.197509	−	0.980301i	$-0.436715\pi$
$101$	3.96989	0.395019	0.197509	+	0.980301i	$0.436715\pi$
$102$	0	0
$103$	−4.23506	−0.417293	−0.208647	−	0.977991i	$-0.566906\pi$
$103$	−4.23506	−0.417293	−0.208647	+	0.977991i	$0.566906\pi$
$104$	−3.14764	−0.308652
$105$	0	0
$106$	−0.204952	−0.0199067
$107$	11.8392	1.14454	0.572271	−	0.820065i	$-0.306063\pi$
$107$	11.8392	1.14454	0.572271	+	0.820065i	$0.306063\pi$
$108$	10.1268	0.974449
$109$	12.0227	1.15157	0.575785	−	0.817601i	$-0.304697\pi$
$109$	12.0227	1.15157	0.575785	+	0.817601i	$0.304697\pi$
$110$	0	0
$111$	−6.96343	−0.660940
$112$	12.3720	1.16904
$113$	10.7915	1.01518	0.507588	−	0.861600i	$-0.330537\pi$
$113$	10.7915	1.01518	0.507588	+	0.861600i	$0.330537\pi$
$114$	−1.49515	−0.140034
$115$	0	0
$116$	−10.4760	−0.972675
$117$	−4.50961	−0.416913
$118$	−1.85236	−0.170523
$119$	0	0
$120$	0	0
$121$	22.0049	2.00045
$122$	2.72641	0.246838
$123$	4.44293	0.400605
$124$	−15.8718	−1.42533
$125$	0	0
$126$	−1.95306	−0.173992
$127$	17.1383	1.52078	0.760388	−	0.649469i	$-0.225009\pi$
$127$	17.1383	1.52078	0.760388	+	0.649469i	$0.225009\pi$
$128$	−8.78568	−0.776552
$129$	−10.9572	−0.964730
$130$	0	0
$131$	8.89551	0.777204	0.388602	−	0.921406i	$-0.372958\pi$
$131$	8.89551	0.777204	0.388602	+	0.921406i	$0.372958\pi$
$132$	12.2745	1.06836
$133$	−15.4477	−1.33948
$134$	−0.133353	−0.0115200
$135$	0	0
$136$	0	0
$137$	−14.0716	−1.20222	−0.601109	−	0.799167i	$-0.705274\pi$
$137$	−14.0716	−1.20222	−0.601109	+	0.799167i	$0.705274\pi$
$138$	0.514052	0.0437590
$139$	−16.7876	−1.42390	−0.711952	−	0.702228i	$-0.752189\pi$
$139$	−16.7876	−1.42390	−0.711952	+	0.702228i	$0.752189\pi$
$140$	0	0
$141$	7.43416	0.626070
$142$	0.424104	0.0355901
$143$	−14.8916	−1.24530
$144$	−5.96497	−0.497081
$145$	0	0
$146$	−1.01395	−0.0839155
$147$	6.75895	0.557468
$148$	11.8054	0.970400
$149$	−4.76494	−0.390359	−0.195179	−	0.980768i	$-0.562529\pi$
$149$	−4.76494	−0.390359	−0.195179	+	0.980768i	$0.562529\pi$
$150$	0	0
$151$	−2.99063	−0.243374	−0.121687	−	0.992569i	$-0.538830\pi$
$151$	−2.99063	−0.243374	−0.121687	+	0.992569i	$0.538830\pi$
$152$	5.19850	0.421654
$153$	0	0
$154$	−6.44938	−0.519706
$155$	0	0
$156$	−5.53820	−0.443411
$157$	13.9447	1.11291	0.556454	−	0.830878i	$-0.312162\pi$
$157$	13.9447	1.11291	0.556454	+	0.830878i	$0.312162\pi$
$158$	0.589850	0.0469260
$159$	−0.739554	−0.0586504
$160$	0	0
$161$	5.31111	0.418574
$162$	−0.234587	−0.0184309
$163$	−8.65491	−0.677905	−0.338953	−	0.940803i	$-0.610073\pi$
$163$	−8.65491	−0.677905	−0.338953	+	0.940803i	$0.610073\pi$
$164$	−7.53230	−0.588174
$165$	0	0
$166$	5.18421	0.402373
$167$	0.773357	0.0598442	0.0299221	−	0.999552i	$-0.490474\pi$
$167$	0.773357	0.0598442	0.0299221	+	0.999552i	$0.490474\pi$
$168$	−4.91903	−0.379512
$169$	−6.28100	−0.483154
$170$	0	0
$171$	7.44785	0.569551
$172$	18.5763	1.41643
$173$	−2.41097	−0.183302	−0.0916511	−	0.995791i	$-0.529214\pi$
$173$	−2.41097	−0.183302	−0.0916511	+	0.995791i	$0.529214\pi$
$174$	1.92242	0.145739
$175$	0	0
$176$	−19.6975	−1.48476
$177$	−6.68409	−0.502407
$178$	−1.09679	−0.0822077
$179$	18.4558	1.37945	0.689727	−	0.724070i	$-0.257731\pi$
$179$	18.4558	1.37945	0.689727	+	0.724070i	$0.257731\pi$
$180$	0	0
$181$	20.6366	1.53391	0.766953	−	0.641703i	$-0.221772\pi$
$181$	20.6366	1.53391	0.766953	+	0.641703i	$0.221772\pi$
$182$	2.90992	0.215698
$183$	9.83807	0.727251
$184$	−1.78731	−0.131762
$185$	0	0
$186$	2.91258	0.213561
$187$	0	0
$188$	−12.6035	−0.919203
$189$	−19.2000	−1.39660
$190$	0	0
$191$	22.8113	1.65057	0.825286	−	0.564716i	$-0.191014\pi$
$191$	22.8113	1.65057	0.825286	+	0.564716i	$0.191014\pi$
$192$	−6.47730	−0.467459
$193$	−19.7208	−1.41953	−0.709767	−	0.704437i	$-0.751200\pi$
$193$	−19.7208	−1.41953	−0.709767	+	0.704437i	$0.751200\pi$
$194$	3.64223	0.261497
$195$	0	0
$196$	−11.4588	−0.818482
$197$	12.8870	0.918160	0.459080	−	0.888395i	$-0.348179\pi$
$197$	12.8870	0.918160	0.459080	+	0.888395i	$0.348179\pi$
$198$	3.10947	0.220980
$199$	6.44350	0.456767	0.228384	−	0.973571i	$-0.426656\pi$
$199$	6.44350	0.456767	0.228384	+	0.973571i	$0.426656\pi$
$200$	0	0
$201$	−0.481195	−0.0339409
$202$	1.23506	0.0868988
$203$	19.8622	1.39405
$204$	0	0
$205$	0	0
$206$	−1.31756	−0.0917988
$207$	−2.56067	−0.177979
$208$	8.88739	0.616230
$209$	24.5943	1.70122
$210$	0	0
$211$	−11.3580	−0.781920	−0.390960	−	0.920408i	$-0.627857\pi$
$211$	−11.3580	−0.781920	−0.390960	+	0.920408i	$0.627857\pi$
$212$	1.25380	0.0861113
$213$	1.53035	0.104858
$214$	3.68328	0.251784
$215$	0	0
$216$	6.46126	0.439633
$217$	30.0923	2.04280
$218$	3.74037	0.253330
$219$	−3.65878	−0.247237
$220$	0	0
$221$	0	0
$222$	−2.16638	−0.145398
$223$	0.888922	0.0595266	0.0297633	−	0.999557i	$-0.490525\pi$
$223$	0.888922	0.0595266	0.0297633	+	0.999557i	$0.490525\pi$
$224$	12.6126	0.842714
$225$	0	0
$226$	3.35731	0.223325
$227$	12.9280	0.858064	0.429032	−	0.903289i	$-0.358855\pi$
$227$	12.9280	0.858064	0.429032	+	0.903289i	$0.358855\pi$
$228$	9.14662	0.605750
$229$	−2.83161	−0.187118	−0.0935591	−	0.995614i	$-0.529824\pi$
$229$	−2.83161	−0.187118	−0.0935591	+	0.995614i	$0.529824\pi$
$230$	0	0
$231$	−23.2721	−1.53119
$232$	−6.68409	−0.438832
$233$	−19.9381	−1.30619	−0.653094	−	0.757277i	$-0.726529\pi$
$233$	−19.9381	−1.30619	−0.653094	+	0.757277i	$0.726529\pi$
$234$	−1.40297	−0.0917153
$235$	0	0
$236$	11.3319	0.737641
$237$	2.12843	0.138256
$238$	0	0
$239$	27.2859	1.76498	0.882490	−	0.470332i	$-0.155866\pi$
$239$	27.2859	1.76498	0.882490	+	0.470332i	$0.155866\pi$
$240$	0	0
$241$	−3.57462	−0.230262	−0.115131	−	0.993350i	$-0.536729\pi$
$241$	−3.57462	−0.230262	−0.115131	+	0.993350i	$0.536729\pi$
$242$	6.84590	0.440071
$243$	15.1162	0.969703
$244$	−16.6789	−1.06776
$245$	0	0
$246$	1.38223	0.0881278
$247$	−11.0968	−0.706072
$248$	−10.1268	−0.643051
$249$	18.7068	1.18550
$250$	0	0
$251$	17.6128	1.11171	0.555857	−	0.831278i	$-0.312390\pi$
$251$	17.6128	1.11171	0.555857	+	0.831278i	$0.312390\pi$
$252$	11.9479	0.752646
$253$	−8.45584	−0.531614
$254$	5.33185	0.334550
$255$	0	0
$256$	8.80642	0.550401
$257$	20.5462	1.28163	0.640817	−	0.767693i	$-0.278596\pi$
$257$	20.5462	1.28163	0.640817	+	0.767693i	$0.278596\pi$
$258$	−3.40888	−0.212227
$259$	−22.3827	−1.39079
$260$	0	0
$261$	−9.57625	−0.592755
$262$	2.76746	0.170974
$263$	8.38715	0.517174	0.258587	−	0.965988i	$-0.416743\pi$
$263$	8.38715	0.517174	0.258587	+	0.965988i	$0.416743\pi$
$264$	7.83161	0.482002
$265$	0	0
$266$	−4.80589	−0.294668
$267$	−3.95768	−0.242206
$268$	0.815792	0.0498324
$269$	−12.4058	−0.756395	−0.378197	−	0.925725i	$-0.623456\pi$
$269$	−12.4058	−0.756395	−0.378197	+	0.925725i	$0.623456\pi$
$270$	0	0
$271$	−7.13828	−0.433619	−0.216810	−	0.976214i	$-0.569565\pi$
$271$	−7.13828	−0.433619	−0.216810	+	0.976214i	$0.569565\pi$
$272$	0	0
$273$	10.5002	0.635503
$274$	−4.37778	−0.264472
$275$	0	0
$276$	−3.14473	−0.189290
$277$	−20.4193	−1.22688	−0.613438	−	0.789743i	$-0.710214\pi$
$277$	−20.4193	−1.22688	−0.613438	+	0.789743i	$0.710214\pi$
$278$	−5.22275	−0.313240
$279$	−14.5086	−0.868605
$280$	0	0
$281$	−29.1432	−1.73854	−0.869269	−	0.494340i	$-0.835410\pi$
$281$	−29.1432	−1.73854	−0.869269	+	0.494340i	$0.835410\pi$
$282$	2.31283	0.137727
$283$	−9.16991	−0.545095	−0.272547	−	0.962142i	$-0.587866\pi$
$283$	−9.16991	−0.545095	−0.272547	+	0.962142i	$0.587866\pi$
$284$	−2.59447	−0.153954
$285$	0	0
$286$	−4.63290	−0.273949
$287$	14.2810	0.842981
$288$	−6.08097	−0.358324
$289$	0	0
$290$	0	0
$291$	13.1427	0.770440
$292$	6.20290	0.362997
$293$	16.2351	0.948463	0.474231	−	0.880400i	$-0.342726\pi$
$293$	16.2351	0.948463	0.474231	+	0.880400i	$0.342726\pi$
$294$	2.10276	0.122636
$295$	0	0
$296$	7.53230	0.437806
$297$	30.5684	1.77376
$298$	−1.48241	−0.0858737
$299$	3.81522	0.220640
$300$	0	0
$301$	−35.2200	−2.03005
$302$	−0.930409	−0.0535390
$303$	4.45664	0.256027
$304$	−14.6780	−0.841841
$305$	0	0
$306$	0	0
$307$	−3.37778	−0.192780	−0.0963902	−	0.995344i	$-0.530730\pi$
$307$	−3.37778	−0.192780	−0.0963902	+	0.995344i	$0.530730\pi$
$308$	39.4543	2.24812
$309$	−4.75432	−0.270464
$310$	0	0
$311$	7.95641	0.451166	0.225583	−	0.974224i	$-0.427571\pi$
$311$	7.95641	0.451166	0.225583	+	0.974224i	$0.427571\pi$
$312$	−3.53358	−0.200049
$313$	6.83380	0.386269	0.193135	−	0.981172i	$-0.438135\pi$
$313$	6.83380	0.386269	0.193135	+	0.981172i	$0.438135\pi$
$314$	4.33830	0.244825
$315$	0	0
$316$	−3.60843	−0.202990
$317$	−27.1550	−1.52517	−0.762587	−	0.646886i	$-0.776071\pi$
$317$	−27.1550	−1.52517	−0.762587	+	0.646886i	$0.776071\pi$
$318$	−0.230081	−0.0129023
$319$	−31.6227	−1.77053
$320$	0	0
$321$	13.2908	0.741822
$322$	1.65233	0.0920806
$323$	0	0
$324$	1.43509	0.0797274
$325$	0	0
$326$	−2.69261	−0.149130
$327$	13.4968	0.746377
$328$	−4.80589	−0.265361
$329$	23.8958	1.31742
$330$	0	0
$331$	20.3783	1.12009	0.560045	−	0.828462i	$-0.310784\pi$
$331$	20.3783	1.12009	0.560045	+	0.828462i	$0.310784\pi$
$332$	−31.7146	−1.74056
$333$	10.7915	0.591369
$334$	0.240597	0.0131649
$335$	0	0
$336$	13.8889	0.757703
$337$	−4.70775	−0.256447	−0.128224	−	0.991745i	$-0.540928\pi$
$337$	−4.70775	−0.256447	−0.128224	+	0.991745i	$0.540928\pi$
$338$	−1.95407	−0.106287
$339$	12.1146	0.657976
$340$	0	0
$341$	−47.9101	−2.59448
$342$	2.31708	0.125294
$343$	−3.53358	−0.190795
$344$	11.8524	0.639036
$345$	0	0
$346$	−0.750070	−0.0403240
$347$	35.3520	1.89779	0.948896	−	0.315588i	$-0.102202\pi$
$347$	35.3520	1.89779	0.948896	+	0.315588i	$0.102202\pi$
$348$	−11.7605	−0.630428
$349$	25.0558	1.34121	0.670603	−	0.741817i	$-0.266036\pi$
$349$	25.0558	1.34121	0.670603	+	0.741817i	$0.266036\pi$
$350$	0	0
$351$	−13.7923	−0.736178
$352$	−20.0806	−1.07030
$353$	−13.6953	−0.728930	−0.364465	−	0.931217i	$-0.618748\pi$
$353$	−13.6953	−0.728930	−0.364465	+	0.931217i	$0.618748\pi$
$354$	−2.07947	−0.110523
$355$	0	0
$356$	6.70964	0.355610
$357$	0	0
$358$	5.74176	0.303461
$359$	−2.53480	−0.133781	−0.0668907	−	0.997760i	$-0.521308\pi$
$359$	−2.53480	−0.133781	−0.0668907	+	0.997760i	$0.521308\pi$
$360$	0	0
$361$	−0.673071	−0.0354248
$362$	6.42021	0.337439
$363$	24.7029	1.29657
$364$	−17.8015	−0.933054
$365$	0	0
$366$	3.06070	0.159985
$367$	−36.3426	−1.89707	−0.948535	−	0.316673i	$-0.897434\pi$
$367$	−36.3426	−1.89707	−0.948535	+	0.316673i	$0.897434\pi$
$368$	5.04649	0.263066
$369$	−6.88536	−0.358438
$370$	0	0
$371$	−2.37716	−0.123416
$372$	−17.8178	−0.923810
$373$	12.7714	0.661278	0.330639	−	0.943757i	$-0.392736\pi$
$373$	12.7714	0.661278	0.330639	+	0.943757i	$0.392736\pi$
$374$	0	0
$375$	0	0
$376$	−8.04149	−0.414708
$377$	14.2680	0.734837
$378$	−5.97328	−0.307232
$379$	−22.1495	−1.13774	−0.568872	−	0.822426i	$-0.692620\pi$
$379$	−22.1495	−1.13774	−0.568872	+	0.822426i	$0.692620\pi$
$380$	0	0
$381$	19.2396	0.985674
$382$	7.09679	0.363103
$383$	−1.95407	−0.0998482	−0.0499241	−	0.998753i	$-0.515898\pi$
$383$	−1.95407	−0.0998482	−0.0499241	+	0.998753i	$0.515898\pi$
$384$	−9.86289	−0.503314
$385$	0	0
$386$	−6.13529	−0.312278
$387$	16.9808	0.863182
$388$	−22.2815	−1.13117
$389$	7.91258	0.401184	0.200592	−	0.979675i	$-0.435713\pi$
$389$	7.91258	0.401184	0.200592	+	0.979675i	$0.435713\pi$
$390$	0	0
$391$	0	0
$392$	−7.31111	−0.369267
$393$	9.98619	0.503736
$394$	4.00924	0.201983
$395$	0	0
$396$	−19.0223	−0.955906
$397$	−9.24476	−0.463981	−0.231991	−	0.972718i	$-0.574524\pi$
$397$	−9.24476	−0.463981	−0.231991	+	0.972718i	$0.574524\pi$
$398$	2.00462	0.100483
$399$	−17.3417	−0.868171
$400$	0	0
$401$	21.4848	1.07290	0.536450	−	0.843932i	$-0.319765\pi$
$401$	21.4848	1.07290	0.536450	+	0.843932i	$0.319765\pi$
$402$	−0.149703	−0.00746653
$403$	21.6168	1.07681
$404$	−7.55554	−0.375902
$405$	0	0
$406$	6.17929	0.306673
$407$	35.6356	1.76639
$408$	0	0
$409$	18.1669	0.898293	0.449147	−	0.893458i	$-0.351728\pi$
$409$	18.1669	0.898293	0.449147	+	0.893458i	$0.351728\pi$
$410$	0	0
$411$	−15.7969	−0.779204
$412$	8.06022	0.397099
$413$	−21.4848	−1.05720
$414$	−0.796644	−0.0391529
$415$	0	0
$416$	9.06022	0.444214
$417$	−18.8459	−0.922888
$418$	7.65147	0.374246
$419$	−14.3589	−0.701476	−0.350738	−	0.936474i	$-0.614069\pi$
$419$	−14.3589	−0.701476	−0.350738	+	0.936474i	$0.614069\pi$
$420$	0	0
$421$	3.96989	0.193481	0.0967403	−	0.995310i	$-0.469158\pi$
$421$	3.96989	0.193481	0.0967403	+	0.995310i	$0.469158\pi$
$422$	−3.53358	−0.172012
$423$	−11.5210	−0.560169
$424$	0.799970	0.0388500
$425$	0	0
$426$	0.476104	0.0230673
$427$	31.6227	1.53033
$428$	−22.5326	−1.08915
$429$	−16.7175	−0.807127
$430$	0	0
$431$	32.9337	1.58636	0.793181	−	0.608985i	$-0.208423\pi$
$431$	32.9337	1.58636	0.793181	+	0.608985i	$0.208423\pi$
$432$	−18.2434	−0.877736
$433$	38.2306	1.83725	0.918623	−	0.395135i	$-0.129302\pi$
$433$	38.2306	1.83725	0.918623	+	0.395135i	$0.129302\pi$
$434$	9.36196	0.449389
$435$	0	0
$436$	−22.8818	−1.09584
$437$	−6.30104	−0.301420
$438$	−1.13828	−0.0543889
$439$	−0.664702	−0.0317245	−0.0158622	−	0.999874i	$-0.505049\pi$
$439$	−0.664702	−0.0317245	−0.0158622	+	0.999874i	$0.505049\pi$
$440$	0	0
$441$	−10.4746	−0.498789
$442$	0	0
$443$	27.5669	1.30974	0.654872	−	0.755740i	$-0.272723\pi$
$443$	27.5669	1.30974	0.654872	+	0.755740i	$0.272723\pi$
$444$	13.2529	0.628954
$445$	0	0
$446$	0.276551	0.0130950
$447$	−5.34916	−0.253007
$448$	−20.8201	−0.983658
$449$	17.4756	0.824723	0.412362	−	0.911020i	$-0.364704\pi$
$449$	17.4756	0.824723	0.412362	+	0.911020i	$0.364704\pi$
$450$	0	0
$451$	−22.7368	−1.07064
$452$	−20.5385	−0.966048
$453$	−3.35731	−0.157740
$454$	4.02201	0.188762
$455$	0	0
$456$	5.83589	0.273290
$457$	−13.0350	−0.609753	−0.304877	−	0.952392i	$-0.598615\pi$
$457$	−13.0350	−0.609753	−0.304877	+	0.952392i	$0.598615\pi$
$458$	−0.880937	−0.0411635
$459$	0	0
$460$	0	0
$461$	10.9491	0.509953	0.254976	−	0.966947i	$-0.417932\pi$
$461$	10.9491	0.509953	0.254976	+	0.966947i	$0.417932\pi$
$462$	−7.24014	−0.336842
$463$	−17.2400	−0.801210	−0.400605	−	0.916251i	$-0.631200\pi$
$463$	−17.2400	−0.801210	−0.400605	+	0.916251i	$0.631200\pi$
$464$	18.8726	0.876138
$465$	0	0
$466$	−6.20290	−0.287344
$467$	−0.0414872	−0.00191980	−0.000959899	−	1.00000i	$-0.500306\pi$
$467$	−0.0414872	−0.00191980	−0.000959899		1.00000i	$0.500306\pi$
$468$	8.58274	0.396737
$469$	−1.54671	−0.0714206
$470$	0	0
$471$	15.6545	0.721319
$472$	7.23014	0.332794
$473$	56.0739	2.57828
$474$	0.662171	0.0304145
$475$	0	0
$476$	0	0
$477$	1.14611	0.0524769
$478$	8.48886	0.388272
$479$	35.4606	1.62024	0.810118	−	0.586266i	$-0.199403\pi$
$479$	35.4606	1.62024	0.810118	+	0.586266i	$0.199403\pi$
$480$	0	0
$481$	−16.0786	−0.733119
$482$	−1.11209	−0.0506545
$483$	5.96230	0.271294
$484$	−41.8800	−1.90364
$485$	0	0
$486$	4.70276	0.213321
$487$	24.6691	1.11787	0.558933	−	0.829213i	$-0.311211\pi$
$487$	24.6691	1.11787	0.558933	+	0.829213i	$0.311211\pi$
$488$	−10.6418	−0.481730
$489$	−9.71609	−0.439377
$490$	0	0
$491$	−8.58073	−0.387243	−0.193621	−	0.981076i	$-0.562023\pi$
$491$	−8.58073	−0.387243	−0.193621	+	0.981076i	$0.562023\pi$
$492$	−8.45584	−0.381219
$493$	0	0
$494$	−3.45230	−0.155326
$495$	0	0
$496$	28.5930	1.28386
$497$	4.91903	0.220649
$498$	5.81984	0.260793
$499$	8.48917	0.380027	0.190014	−	0.981781i	$-0.439147\pi$
$499$	8.48917	0.380027	0.190014	+	0.981781i	$0.439147\pi$
$500$	0	0
$501$	0.868178	0.0387873
$502$	5.47949	0.244562
$503$	24.1879	1.07849	0.539244	−	0.842150i	$-0.318710\pi$
$503$	24.1879	1.07849	0.539244	+	0.842150i	$0.318710\pi$
$504$	7.62320	0.339564
$505$	0	0
$506$	−2.63068	−0.116948
$507$	−7.05111	−0.313151
$508$	−32.6178	−1.44718
$509$	−25.8671	−1.14654	−0.573270	−	0.819366i	$-0.694325\pi$
$509$	−25.8671	−1.14654	−0.573270	+	0.819366i	$0.694325\pi$
$510$	0	0
$511$	−11.7605	−0.520253
$512$	20.3111	0.897633
$513$	22.7787	1.00570
$514$	6.39207	0.281942
$515$	0	0
$516$	20.8539	0.918042
$517$	−38.0446	−1.67320
$518$	−6.96343	−0.305956
$519$	−2.70657	−0.118805
$520$	0	0
$521$	11.4900	0.503385	0.251693	−	0.967807i	$-0.419013\pi$
$521$	11.4900	0.503385	0.251693	+	0.967807i	$0.419013\pi$
$522$	−2.97925	−0.130398
$523$	33.0509	1.44521	0.722606	−	0.691260i	$-0.242944\pi$
$523$	33.0509	1.44521	0.722606	+	0.691260i	$0.242944\pi$
$524$	−16.9300	−0.739592
$525$	0	0
$526$	2.60931	0.113771
$527$	0	0
$528$	−22.1126	−0.962328
$529$	−20.8336	−0.905810
$530$	0	0
$531$	10.3586	0.449524
$532$	29.4002	1.27466
$533$	10.2587	0.444354
$534$	−1.23127	−0.0532820
$535$	0	0
$536$	0.520505	0.0224824
$537$	20.7187	0.894078
$538$	−3.85954	−0.166397
$539$	−34.5891	−1.48986
$540$	0	0
$541$	19.7884	0.850770	0.425385	−	0.905013i	$-0.360139\pi$
$541$	19.7884	0.850770	0.425385	+	0.905013i	$0.360139\pi$
$542$	−2.22077	−0.0953904
$543$	23.1669	0.994185
$544$	0	0
$545$	0	0
$546$	3.26671	0.139802
$547$	−15.4815	−0.661940	−0.330970	−	0.943641i	$-0.607376\pi$
$547$	−15.4815	−0.661940	−0.330970	+	0.943641i	$0.607376\pi$
$548$	26.7812	1.14404
$549$	−15.2464	−0.650700
$550$	0	0
$551$	−23.5643	−1.00387
$552$	−2.00645	−0.0854003
$553$	6.84146	0.290928
$554$	−6.35260	−0.269896
$555$	0	0
$556$	31.9503	1.35500
$557$	−15.3477	−0.650302	−0.325151	−	0.945662i	$-0.605415\pi$
$557$	−15.3477	−0.650302	−0.325151	+	0.945662i	$0.605415\pi$
$558$	−4.51373	−0.191081
$559$	−25.3002	−1.07008
$560$	0	0
$561$	0	0
$562$	−9.06668	−0.382455
$563$	−14.3827	−0.606159	−0.303079	−	0.952965i	$-0.598015\pi$
$563$	−14.3827	−0.606159	−0.303079	+	0.952965i	$0.598015\pi$
$564$	−14.1488	−0.595772
$565$	0	0
$566$	−2.85283	−0.119913
$567$	−2.72089	−0.114267
$568$	−1.65537	−0.0694578
$569$	12.1432	0.509069	0.254535	−	0.967064i	$-0.418078\pi$
$569$	12.1432	0.509069	0.254535	+	0.967064i	$0.418078\pi$
$570$	0	0
$571$	2.88663	0.120802	0.0604009	−	0.998174i	$-0.480762\pi$
$571$	2.88663	0.120802	0.0604009	+	0.998174i	$0.480762\pi$
$572$	28.3419	1.18503
$573$	25.6082	1.06980
$574$	4.44293	0.185444
$575$	0	0
$576$	10.0381	0.418254
$577$	−3.10970	−0.129458	−0.0647291	−	0.997903i	$-0.520618\pi$
$577$	−3.10970	−0.129458	−0.0647291	+	0.997903i	$0.520618\pi$
$578$	0	0
$579$	−22.1388	−0.920055
$580$	0	0
$581$	60.1298	2.49460
$582$	4.08880	0.169486
$583$	3.78469	0.156746
$584$	3.95768	0.163770
$585$	0	0
$586$	5.05086	0.208649
$587$	−10.2034	−0.421140	−0.210570	−	0.977579i	$-0.567532\pi$
$587$	−10.2034	−0.421140	−0.210570	+	0.977579i	$0.567532\pi$
$588$	−12.8637	−0.530490
$589$	−35.7012	−1.47104
$590$	0	0
$591$	14.4671	0.595095
$592$	−21.2675	−0.874089
$593$	16.2301	0.666492	0.333246	−	0.942840i	$-0.391856\pi$
$593$	16.2301	0.666492	0.333246	+	0.942840i	$0.391856\pi$
$594$	9.51008	0.390203
$595$	0	0
$596$	9.06868	0.371468
$597$	7.23353	0.296049
$598$	1.18694	0.0485378
$599$	−8.43309	−0.344567	−0.172283	−	0.985047i	$-0.555114\pi$
$599$	−8.43309	−0.344567	−0.172283	+	0.985047i	$0.555114\pi$
$600$	0	0
$601$	−20.2535	−0.826160	−0.413080	−	0.910695i	$-0.635547\pi$
$601$	−20.2535	−0.826160	−0.413080	+	0.910695i	$0.635547\pi$
$602$	−10.9572	−0.446583
$603$	0.745724	0.0303682
$604$	5.69181	0.231596
$605$	0	0
$606$	1.38649	0.0563225
$607$	22.7732	0.924334	0.462167	−	0.886793i	$-0.347072\pi$
$607$	22.7732	0.924334	0.462167	+	0.886793i	$0.347072\pi$
$608$	−14.9634	−0.606847
$609$	22.2975	0.903540
$610$	0	0
$611$	17.1655	0.694441
$612$	0	0
$613$	−8.40636	−0.339530	−0.169765	−	0.985485i	$-0.554301\pi$
$613$	−8.40636	−0.339530	−0.169765	+	0.985485i	$0.554301\pi$
$614$	−1.05086	−0.0424091
$615$	0	0
$616$	25.1733	1.01426
$617$	14.8168	0.596500	0.298250	−	0.954488i	$-0.403597\pi$
$617$	14.8168	0.596500	0.298250	+	0.954488i	$0.403597\pi$
$618$	−1.47911	−0.0594984
$619$	−31.6221	−1.27100	−0.635500	−	0.772101i	$-0.719206\pi$
$619$	−31.6221	−1.27100	−0.635500	+	0.772101i	$0.719206\pi$
$620$	0	0
$621$	−7.83161	−0.314272
$622$	2.47530	0.0992505
$623$	−12.7212	−0.509666
$624$	9.97707	0.399403
$625$	0	0
$626$	2.12605	0.0849740
$627$	27.6098	1.10263
$628$	−26.5397	−1.05905
$629$	0	0
$630$	0	0
$631$	17.3733	0.691622	0.345811	−	0.938304i	$-0.387604\pi$
$631$	17.3733	0.691622	0.345811	+	0.938304i	$0.387604\pi$
$632$	−2.30231	−0.0915810
$633$	−12.7506	−0.506793
$634$	−8.44812	−0.335518
$635$	0	0
$636$	1.40753	0.0558121
$637$	15.6064	0.618348
$638$	−9.83807	−0.389493
$639$	−2.37164	−0.0938205
$640$	0	0
$641$	−7.26842	−0.287085	−0.143543	−	0.989644i	$-0.545849\pi$
$641$	−7.26842	−0.287085	−0.143543	+	0.989644i	$0.545849\pi$
$642$	4.13488	0.163191
$643$	4.40507	0.173719	0.0868595	−	0.996221i	$-0.472317\pi$
$643$	4.40507	0.173719	0.0868595	+	0.996221i	$0.472317\pi$
$644$	−10.1082	−0.398317
$645$	0	0
$646$	0	0
$647$	0.198022	0.00778504	0.00389252	−	0.999992i	$-0.498761\pi$
$647$	0.198022	0.00778504	0.00389252	+	0.999992i	$0.498761\pi$
$648$	0.915643	0.0359698
$649$	34.2061	1.34271
$650$	0	0
$651$	33.7820	1.32402
$652$	16.4721	0.645098
$653$	−9.92723	−0.388482	−0.194241	−	0.980954i	$-0.562224\pi$
$653$	−9.92723	−0.388482	−0.194241	+	0.980954i	$0.562224\pi$
$654$	4.19897	0.164193
$655$	0	0
$656$	13.5695	0.529798
$657$	5.67014	0.221213
$658$	7.43416	0.289814
$659$	3.00492	0.117055	0.0585276	−	0.998286i	$-0.481359\pi$
$659$	3.00492	0.117055	0.0585276	+	0.998286i	$0.481359\pi$
$660$	0	0
$661$	−23.6923	−0.921523	−0.460762	−	0.887524i	$-0.652424\pi$
$661$	−23.6923	−0.921523	−0.460762	+	0.887524i	$0.652424\pi$
$662$	6.33984	0.246405
$663$	0	0
$664$	−20.2351	−0.785273
$665$	0	0
$666$	3.35731	0.130093
$667$	8.10171	0.313699
$668$	−1.47186	−0.0569481
$669$	0.997912	0.0385815
$670$	0	0
$671$	−50.3466	−1.94361
$672$	14.1590	0.546196
$673$	30.1968	1.16400	0.582001	−	0.813188i	$-0.302270\pi$
$673$	30.1968	1.16400	0.582001	+	0.813188i	$0.302270\pi$
$674$	−1.46462	−0.0564150
$675$	0	0
$676$	11.9541	0.459772
$677$	−47.9413	−1.84253	−0.921266	−	0.388933i	$-0.872844\pi$
$677$	−47.9413	−1.84253	−0.921266	+	0.388933i	$0.872844\pi$
$678$	3.76895	0.144746
$679$	42.2449	1.62121
$680$	0	0
$681$	14.5131	0.556145
$682$	−14.9052	−0.570750
$683$	1.41477	0.0541347	0.0270674	−	0.999634i	$-0.491383\pi$
$683$	1.41477	0.0541347	0.0270674	+	0.999634i	$0.491383\pi$
$684$	−14.1748	−0.541989
$685$	0	0
$686$	−1.09932	−0.0419723
$687$	−3.17880	−0.121279
$688$	−33.4652	−1.27585
$689$	−1.70763	−0.0650554
$690$	0	0
$691$	1.36321	0.0518588	0.0259294	−	0.999664i	$-0.491745\pi$
$691$	1.36321	0.0518588	0.0259294	+	0.999664i	$0.491745\pi$
$692$	4.58858	0.174432
$693$	36.0656	1.37002
$694$	10.9983	0.417489
$695$	0	0
$696$	−7.50363	−0.284424
$697$	0	0
$698$	7.79505	0.295047
$699$	−22.3827	−0.846592
$700$	0	0
$701$	−40.1990	−1.51829	−0.759147	−	0.650919i	$-0.774384\pi$
$701$	−40.1990	−1.51829	−0.759147	+	0.650919i	$0.774384\pi$
$702$	−4.29089	−0.161949
$703$	26.5546	1.00153
$704$	33.1478	1.24930
$705$	0	0
$706$	−4.26073	−0.160355
$707$	14.3251	0.538749
$708$	12.7212	0.478094
$709$	−19.4729	−0.731322	−0.365661	−	0.930748i	$-0.619157\pi$
$709$	−19.4729	−0.731322	−0.365661	+	0.930748i	$0.619157\pi$
$710$	0	0
$711$	−3.29850	−0.123704
$712$	4.28100	0.160437
$713$	12.2745	0.459685
$714$	0	0
$715$	0	0
$716$	−35.1254	−1.31270
$717$	30.6314	1.14395
$718$	−0.788595	−0.0294301
$719$	−21.9765	−0.819586	−0.409793	−	0.912179i	$-0.634399\pi$
$719$	−21.9765	−0.819586	−0.409793	+	0.912179i	$0.634399\pi$
$720$	0	0
$721$	−15.2819	−0.569128
$722$	−0.209398	−0.00779297
$723$	−4.01291	−0.149242
$724$	−39.2758	−1.45967
$725$	0	0
$726$	7.68528	0.285227
$727$	36.7007	1.36116	0.680578	−	0.732676i	$-0.261729\pi$
$727$	36.7007	1.36116	0.680578	+	0.732676i	$0.261729\pi$
$728$	−11.3580	−0.420957
$729$	19.2317	0.712284
$730$	0	0
$731$	0	0
$732$	−18.7239	−0.692056
$733$	−24.8666	−0.918471	−0.459235	−	0.888315i	$-0.651877\pi$
$733$	−24.8666	−0.918471	−0.459235	+	0.888315i	$0.651877\pi$
$734$	−11.3065	−0.417330
$735$	0	0
$736$	5.14462	0.189633
$737$	2.46253	0.0907085
$738$	−2.14209	−0.0788514
$739$	32.7368	1.20424	0.602122	−	0.798404i	$-0.294322\pi$
$739$	32.7368	1.20424	0.602122	+	0.798404i	$0.294322\pi$
$740$	0	0
$741$	−12.4574	−0.457633
$742$	−0.739554	−0.0271499
$743$	21.3762	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$743$	21.3762	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$744$	−11.3684	−0.416786
$745$	0	0
$746$	3.97328	0.145472
$747$	−28.9906	−1.06071
$748$	0	0
$749$	42.7210	1.56099
$750$	0	0
$751$	−44.5557	−1.62586	−0.812930	−	0.582362i	$-0.802129\pi$
$751$	−44.5557	−1.62586	−0.812930	+	0.582362i	$0.802129\pi$
$752$	22.7052	0.827973
$753$	19.7724	0.720545
$754$	4.43887	0.161654
$755$	0	0
$756$	36.5417	1.32901
$757$	−12.1793	−0.442664	−0.221332	−	0.975199i	$-0.571040\pi$
$757$	−12.1793	−0.442664	−0.221332	+	0.975199i	$0.571040\pi$
$758$	−6.89089	−0.250288
$759$	−9.49260	−0.344560
$760$	0	0
$761$	7.18712	0.260533	0.130266	−	0.991479i	$-0.458417\pi$
$761$	7.18712	0.260533	0.130266	+	0.991479i	$0.458417\pi$
$762$	5.98559	0.216835
$763$	43.3832	1.57058
$764$	−43.4148	−1.57069
$765$	0	0
$766$	−0.607926	−0.0219652
$767$	−15.4336	−0.557274
$768$	9.88618	0.356737
$769$	14.9748	0.540005	0.270003	−	0.962860i	$-0.412975\pi$
$769$	14.9748	0.540005	0.270003	+	0.962860i	$0.412975\pi$
$770$	0	0
$771$	23.0653	0.830678
$772$	37.5328	1.35084
$773$	26.2938	0.945721	0.472860	−	0.881137i	$-0.343222\pi$
$773$	26.2938	0.945721	0.472860	+	0.881137i	$0.343222\pi$
$774$	5.28286	0.189888
$775$	0	0
$776$	−14.2164	−0.510339
$777$	−25.1270	−0.901428
$778$	2.46167	0.0882550
$779$	−16.9428	−0.607039
$780$	0	0
$781$	−7.83161	−0.280237
$782$	0	0
$783$	−29.2883	−1.04668
$784$	20.6430	0.737249
$785$	0	0
$786$	3.10678	0.110815
$787$	−9.35342	−0.333413	−0.166707	−	0.986007i	$-0.553313\pi$
$787$	−9.35342	−0.333413	−0.166707	+	0.986007i	$0.553313\pi$
$788$	−24.5267	−0.873727
$789$	9.41550	0.335201
$790$	0	0
$791$	38.9403	1.38456
$792$	−12.1369	−0.431267
$793$	22.7161	0.806672
$794$	−2.87612	−0.102070
$795$	0	0
$796$	−12.2633	−0.434663
$797$	29.4800	1.04423	0.522117	−	0.852874i	$-0.325142\pi$
$797$	29.4800	1.04423	0.522117	+	0.852874i	$0.325142\pi$
$798$	−5.39514	−0.190986
$799$	0	0
$800$	0	0
$801$	6.13335	0.216711
$802$	6.68409	0.236024
$803$	18.7239	0.660753
$804$	0.915816	0.0322983
$805$	0	0
$806$	6.72514	0.236883
$807$	−13.9269	−0.490249
$808$	−4.82071	−0.169592
$809$	35.1219	1.23482	0.617410	−	0.786642i	$-0.288182\pi$
$809$	35.1219	1.23482	0.617410	+	0.786642i	$0.288182\pi$
$810$	0	0
$811$	−17.5682	−0.616902	−0.308451	−	0.951240i	$-0.599811\pi$
$811$	−17.5682	−0.616902	−0.308451	+	0.951240i	$0.599811\pi$
$812$	−37.8020	−1.32659
$813$	−8.01350	−0.281046
$814$	11.0865	0.388582
$815$	0	0
$816$	0	0
$817$	41.7846	1.46186
$818$	5.65185	0.197612
$819$	−16.2726	−0.568610
$820$	0	0
$821$	−13.1043	−0.457343	−0.228672	−	0.973504i	$-0.573438\pi$
$821$	−13.1043	−0.457343	−0.228672	+	0.973504i	$0.573438\pi$
$822$	−4.91454	−0.171414
$823$	23.7638	0.828355	0.414178	−	0.910196i	$-0.364069\pi$
$823$	23.7638	0.828355	0.414178	+	0.910196i	$0.364069\pi$
$824$	5.14272	0.179155
$825$	0	0
$826$	−6.68409	−0.232569
$827$	−28.9423	−1.00642	−0.503211	−	0.864164i	$-0.667848\pi$
$827$	−28.9423	−1.00642	−0.503211	+	0.864164i	$0.667848\pi$
$828$	4.87350	0.169366
$829$	48.3131	1.67798	0.838992	−	0.544144i	$-0.183145\pi$
$829$	48.3131	1.67798	0.838992	+	0.544144i	$0.183145\pi$
$830$	0	0
$831$	−22.9229	−0.795187
$832$	−14.9561	−0.518509
$833$	0	0
$834$	−5.86311	−0.203023
$835$	0	0
$836$	−46.8081	−1.61889
$837$	−44.3733	−1.53377
$838$	−4.46715	−0.154315
$839$	−36.8238	−1.27130	−0.635650	−	0.771978i	$-0.719268\pi$
$839$	−36.8238	−1.27130	−0.635650	+	0.771978i	$0.719268\pi$
$840$	0	0
$841$	1.29835	0.0447707
$842$	1.23506	0.0425631
$843$	−32.7164	−1.12681
$844$	21.6168	0.744079
$845$	0	0
$846$	−3.58427	−0.123230
$847$	79.4031	2.72832
$848$	−2.25872	−0.0775648
$849$	−10.2942	−0.353297
$850$	0	0
$851$	−9.12981	−0.312966
$852$	−2.91258	−0.0997833
$853$	−29.8138	−1.02080	−0.510402	−	0.859936i	$-0.670503\pi$
$853$	−29.8138	−1.02080	−0.510402	+	0.859936i	$0.670503\pi$
$854$	9.83807	0.336652
$855$	0	0
$856$	−14.3766	−0.491383
$857$	−1.71246	−0.0584965	−0.0292483	−	0.999572i	$-0.509311\pi$
$857$	−1.71246	−0.0584965	−0.0292483	+	0.999572i	$0.509311\pi$
$858$	−5.20094	−0.177557
$859$	−22.5749	−0.770246	−0.385123	−	0.922865i	$-0.625841\pi$
$859$	−22.5749	−0.770246	−0.385123	+	0.922865i	$0.625841\pi$
$860$	0	0
$861$	16.0320	0.546369
$862$	10.2459	0.348978
$863$	−54.2578	−1.84696	−0.923479	−	0.383650i	$-0.874667\pi$
$863$	−54.2578	−1.84696	−0.923479	+	0.383650i	$0.874667\pi$
$864$	−18.5982	−0.632723
$865$	0	0
$866$	11.8938	0.404169
$867$	0	0
$868$	−57.2721	−1.94394
$869$	−10.8923	−0.369496
$870$	0	0
$871$	−1.11108	−0.0376474
$872$	−14.5995	−0.494400
$873$	−20.3677	−0.689343
$874$	−1.96030	−0.0663082
$875$	0	0
$876$	6.96343	0.235273
$877$	−14.8989	−0.503099	−0.251549	−	0.967844i	$-0.580940\pi$
$877$	−14.8989	−0.503099	−0.251549	+	0.967844i	$0.580940\pi$
$878$	−0.206794	−0.00697896
$879$	18.2256	0.614736
$880$	0	0
$881$	−22.3330	−0.752419	−0.376209	−	0.926535i	$-0.622773\pi$
$881$	−22.3330	−0.752419	−0.376209	+	0.926535i	$0.622773\pi$
$882$	−3.25872	−0.109727
$883$	−25.3319	−0.852485	−0.426242	−	0.904609i	$-0.640163\pi$
$883$	−25.3319	−0.852485	−0.426242	+	0.904609i	$0.640163\pi$
$884$	0	0
$885$	0	0
$886$	8.57628	0.288126
$887$	6.35813	0.213485	0.106743	−	0.994287i	$-0.465958\pi$
$887$	6.35813	0.213485	0.106743	+	0.994287i	$0.465958\pi$
$888$	8.45584	0.283759
$889$	61.8422	2.07412
$890$	0	0
$891$	4.33194	0.145125
$892$	−1.69181	−0.0566459
$893$	−28.3497	−0.948686
$894$	−1.66417	−0.0556581
$895$	0	0
$896$	−31.7025	−1.05911
$897$	4.28300	0.143005
$898$	5.43679	0.181428
$899$	45.9037	1.53097
$900$	0	0
$901$	0	0
$902$	−7.07361	−0.235525
$903$	−39.5383	−1.31575
$904$	−13.1043	−0.435843
$905$	0	0
$906$	−1.04449	−0.0347007
$907$	−13.0295	−0.432636	−0.216318	−	0.976323i	$-0.569405\pi$
$907$	−13.0295	−0.432636	−0.216318	+	0.976323i	$0.569405\pi$
$908$	−24.6048	−0.816539
$909$	−6.90660	−0.229078
$910$	0	0
$911$	−20.7685	−0.688093	−0.344046	−	0.938953i	$-0.611798\pi$
$911$	−20.7685	−0.688093	−0.344046	+	0.938953i	$0.611798\pi$
$912$	−16.4777	−0.545630
$913$	−95.7328	−3.16829
$914$	−4.05530	−0.134137
$915$	0	0
$916$	5.38916	0.178063
$917$	32.0988	1.06000
$918$	0	0
$919$	−5.10970	−0.168553	−0.0842766	−	0.996442i	$-0.526858\pi$
$919$	−5.10970	−0.168553	−0.0842766	+	0.996442i	$0.526858\pi$
$920$	0	0
$921$	−3.79193	−0.124948
$922$	3.40636	0.112183
$923$	3.53358	0.116309
$924$	44.2918	1.45709
$925$	0	0
$926$	−5.36349	−0.176255
$927$	7.36794	0.241995
$928$	19.2396	0.631571
$929$	39.6073	1.29947	0.649737	−	0.760159i	$-0.274879\pi$
$929$	39.6073	1.29947	0.649737	+	0.760159i	$0.274879\pi$
$930$	0	0
$931$	−25.7748	−0.844734
$932$	37.9464	1.24298
$933$	8.93194	0.292419
$934$	−0.0129070	−0.000422330 0
$935$	0	0
$936$	5.47610	0.178992
$937$	−39.5575	−1.29229	−0.646144	−	0.763215i	$-0.723620\pi$
$937$	−39.5575	−1.29229	−0.646144	+	0.763215i	$0.723620\pi$
$938$	−0.481195	−0.0157116
$939$	7.67169	0.250356
$940$	0	0
$941$	−53.2283	−1.73519	−0.867597	−	0.497268i	$-0.834337\pi$
$941$	−53.2283	−1.73519	−0.867597	+	0.497268i	$0.834337\pi$
$942$	4.87022	0.158680
$943$	5.82516	0.189693
$944$	−20.4143	−0.664430
$945$	0	0
$946$	17.4450	0.567187
$947$	0.773357	0.0251307	0.0125654	−	0.999921i	$-0.496000\pi$
$947$	0.773357	0.0251307	0.0125654	+	0.999921i	$0.496000\pi$
$948$	−4.05086	−0.131566
$949$	−8.44812	−0.274238
$950$	0	0
$951$	−30.4844	−0.988525
$952$	0	0
$953$	36.8731	1.19444	0.597218	−	0.802079i	$-0.296273\pi$
$953$	36.8731	1.19444	0.597218	+	0.802079i	$0.296273\pi$
$954$	0.356564	0.0115442
$955$	0	0
$956$	−51.9309	−1.67956
$957$	−35.4999	−1.14755
$958$	11.0321	0.356430
$959$	−50.7763	−1.63965
$960$	0	0
$961$	38.5466	1.24344
$962$	−5.00217	−0.161276
$963$	−20.5973	−0.663738
$964$	6.80327	0.219118
$965$	0	0
$966$	1.85492	0.0596810
$967$	−17.8306	−0.573392	−0.286696	−	0.958022i	$-0.592557\pi$
$967$	−17.8306	−0.573392	−0.286696	+	0.958022i	$0.592557\pi$
$968$	−26.7210	−0.858846
$969$	0	0
$970$	0	0
$971$	20.3783	0.653970	0.326985	−	0.945030i	$-0.393967\pi$
$971$	20.3783	0.653970	0.326985	+	0.945030i	$0.393967\pi$
$972$	−28.7693	−0.922775
$973$	−60.5768	−1.94200
$974$	7.67476	0.245915
$975$	0	0
$976$	30.0471	0.961785
$977$	8.64587	0.276606	0.138303	−	0.990390i	$-0.455835\pi$
$977$	8.64587	0.276606	0.138303	+	0.990390i	$0.455835\pi$
$978$	−3.02275	−0.0966569
$979$	20.2535	0.647306
$980$	0	0
$981$	−20.9165	−0.667813
$982$	−2.66953	−0.0851882
$983$	−61.1615	−1.95075	−0.975374	−	0.220558i	$-0.929212\pi$
$983$	−61.1615	−1.95075	−0.975374	+	0.220558i	$0.929212\pi$
$984$	−5.39514	−0.171991
$985$	0	0
$986$	0	0
$987$	26.8256	0.853869
$988$	21.1195	0.671902
$989$	−14.3661	−0.456815
$990$	0	0
$991$	−27.0979	−0.860792	−0.430396	−	0.902640i	$-0.641626\pi$
$991$	−27.0979	−0.860792	−0.430396	+	0.902640i	$0.641626\pi$
$992$	29.1491	0.925484
$993$	22.8768	0.725974
$994$	1.53035	0.0485397
$995$	0	0
$996$	−35.6031	−1.12813
$997$	−48.3759	−1.53208	−0.766040	−	0.642793i	$-0.777775\pi$
$997$	−48.3759	−1.53208	−0.766040	+	0.642793i	$0.777775\pi$
$998$	2.64105	0.0836009
$999$	33.0049	1.04423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bg.1.4		6
5.4	even	2		7225.2.a.ba.1.3		6
17.4	even	4		425.2.d.a.101.3	✓	6
17.13	even	4		425.2.d.a.101.4	yes	6
17.16	even	2	inner	7225.2.a.bg.1.3		6
85.4	even	4		425.2.d.b.101.4	yes	6
85.13	odd	4		425.2.c.c.424.7		12
85.38	odd	4		425.2.c.c.424.8		12
85.47	odd	4		425.2.c.c.424.6		12
85.64	even	4		425.2.d.b.101.3	yes	6
85.72	odd	4		425.2.c.c.424.5		12
85.84	even	2		7225.2.a.ba.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.c.c.424.5		12	85.72	odd	4
425.2.c.c.424.6		12	85.47	odd	4
425.2.c.c.424.7		12	85.13	odd	4
425.2.c.c.424.8		12	85.38	odd	4
425.2.d.a.101.3	✓	6	17.4	even	4
425.2.d.a.101.4	yes	6	17.13	even	4
425.2.d.b.101.3	yes	6	85.64	even	4
425.2.d.b.101.4	yes	6	85.4	even	4
7225.2.a.ba.1.3		6	5.4	even	2
7225.2.a.ba.1.4		6	85.84	even	2
7225.2.a.bg.1.3		6	17.16	even	2	inner
7225.2.a.bg.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} +1.12261 q^{3} -1.90321 q^{4} +0.349253 q^{6} +3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} -5.74499 q^{11} -2.13656 q^{12} +2.59210 q^{13} +1.12261 q^{14} +3.42864 q^{16} -0.541249 q^{18} -4.28100 q^{19} +4.05086 q^{21} -1.78731 q^{22} +1.47186 q^{23} -1.36321 q^{24} +0.806424 q^{26} -5.32089 q^{27} -6.86760 q^{28} +5.50439 q^{29} +8.33946 q^{31} +3.49532 q^{32} -6.44938 q^{33} +3.31111 q^{36} -6.20290 q^{37} -1.33185 q^{38} +2.90992 q^{39} +3.95768 q^{41} +1.26025 q^{42} -9.76049 q^{43} +10.9339 q^{44} +0.457908 q^{46} +6.62222 q^{47} +3.84902 q^{48} +6.02074 q^{49} -4.93332 q^{52} -0.658781 q^{53} -1.65537 q^{54} -4.38178 q^{56} -4.80589 q^{57} +1.71246 q^{58} -5.95407 q^{59} +8.76357 q^{61} +2.59447 q^{62} -6.27775 q^{63} -5.76986 q^{64} -2.00645 q^{66} -0.428639 q^{67} +1.65233 q^{69} +1.36321 q^{71} +2.11261 q^{72} -3.25917 q^{73} -1.92977 q^{74} +8.14764 q^{76} -20.7304 q^{77} +0.905299 q^{78} +1.89597 q^{79} -0.754037 q^{81} +1.23127 q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} +6.97626 q^{88} -3.52543 q^{89} +9.35342 q^{91} -2.80127 q^{92} +9.36196 q^{93} +2.06022 q^{94} +3.92388 q^{96} +11.7073 q^{97} +1.87310 q^{98} +9.99483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9} + 2 q^{13} - 6 q^{16} - 16 q^{18} - 12 q^{19} - 2 q^{21} - 22 q^{26} - 6 q^{32} + 28 q^{33} + 20 q^{36} + 32 q^{38} + 34 q^{42} + 8 q^{43} + 40 q^{47} - 4 q^{49}+ \cdots - 16 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 16 * q^9 + 2 * q^13 - 6 * q^16 - 16 * q^18 - 12 * q^19 - 2 * q^21 - 22 * q^26 - 6 * q^32 + 28 * q^33 + 20 * q^36 + 32 * q^38 + 34 * q^42 + 8 * q^43 + 40 * q^47 - 4 * q^49 - 30 * q^52 + 10 * q^53 + 4 * q^59 - 22 * q^64 + 16 * q^66 + 24 * q^67 + 24 * q^69 + 52 * q^72 + 36 * q^76 - 44 * q^77 + 34 * q^81 + 20 * q^83 - 6 * q^84 - 4 * q^86 + 76 * q^87 - 8 * q^89 + 30 * q^93 + 40 * q^94 - 16 * q^98

Introduction

L-functions

Modular forms

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